Wild models of curves

Lorenzini, Dino

doi:10.2140/ant.2014.8.331

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Wild models of curves

Dino Lorenzini

Vol. 8 (2014), No. 2, 331–367

DOI: 10.2140/ant.2014.8.331

Abstract

Let $K$ be a complete discrete valuation field with ring of integers $O_{K}$ and algebraically closed residue field $k$ of characteristic $p > 0$ . Let $X ∕ K$ be a smooth proper geometrically connected curve of genus $g > 0$ with $X (K) \neq \emptyset$ if $g = 1$ . Assume that $X ∕ K$ does not have good reduction and that it obtains good reduction over a Galois extension $L ∕ K$ of degree $p$ . Let $Y ∕ O_{L}$ be the smooth model of $X_{L} ∕ L$ . Let $H : = Gal (L ∕ K)$ .

In this article, we provide information on the regular model of $X ∕ K$ obtained by desingularizing the wild quotient singularities of the quotient $Y ∕ H$ . The most precise information on the resolution of these quotient singularities is obtained when the special fiber $Y_{k} ∕ k$ is ordinary. As a corollary, we are able to produce for each odd prime $p$ an infinite class of wild quotient singularities having pairwise distinct resolution graphs. The information on the regular model of $X ∕ K$ also allows us to gather insight into the $p$ -part of the component group of the Néron model of the Jacobian of $X$ .

Keywords

model of a curve, ordinary curve, cyclic quotient singularity, wild ramification, arithmetical tree, resolution graph, component group, Néron model

Mathematical Subject Classification 2010

Primary: 14G20

Secondary: 14G17, 14K15, 14J17

Milestones

Received: 3 January 2013

Revised: 6 June 2013

Accepted: 16 July 2013

Published: 18 May 2014

Authors

Dino Lorenzini
Department of Mathematics University of Georgia Athens, GA 30602 United States

Vol. 8, No. 2, 2014